Why do bones without visible weights move the mesh?
Actually blue doesn't necessarily mean zero weight. To make sure that a bone has zero influence you need to enable the Zero Weight option in the Viewport Overlays panel. The black areas mean real zero weight. Here you can see that the second bone has influence on a large part of your object.
I don't know why zero weight is not displayed by default in Weight Paint mode by the way, blue confuses people as they may think there's no influence when actually there might be.

Also note that you can make sure that a bone has no influence: select it and in the Properties panel &gt; Bone, disable the Deform option.

The first bone's Deform option is deactivated but as it makes the others rotate, and as the others have influence on the top of the mesh, rotating the first bone will move the top of the mesh.
I guess you need to set your armature correctly, reparent the mesh to the armature, and give corrections in Weight Paint mode (most of the time parenting With Automatic Weight is not right away correct).